j. funct. biomater. open access journal of functional ...e-mail: [email protected] 3...

J. Funct. Biomater. 2015, 6, 585-597; doi:10.3390/jfb6030585OPEN ACCESS

Journal ofFunctional

BiomaterialsISSN 2079-4983

www.mdpi.com/journal/jfbArticle

An Efficient Finite Element Framework to Assess FlexibilityPerformances of SMA Self-Expandable Carotid Artery StentsMauro Ferraro 1,*, Ferdinando Auricchio 1 , Elisa Boatti 1, Giulia Scalet 2, Michele Conti 1,Simone Morganti 3 and Alessandro Reali 1,4

1 Department of Civil Engineering and Architecture (DICAR), Università di Pavia , Pavia 27100, Italy;E-Mails: [email protected] (F.A.); [email protected] (E.B.); [email protected] (M.C.);[email protected] (A.R.)

2 Laboratoire de Mécanique des Solides, École Polytechnique, Palaiseau 91128, France;E-Mail: [email protected]

3 Department of Electrical, Computer and Biomedical Engineering (DIII) , Università di Pavia,Pavia 27100, Italy; E-Mail: [email protected]

4 Institute for Advanced Study, Technische Universität München, Lichtenbergstraße 2a,85748 Garching, Germany

* Author to whom correspondence should be addressed; E-Mail: [email protected].

Academic Editor: Lorenza Petrini

Received: 28 April 2015 / Accepted: 10 July 2015 / Published: 14 July 2015

Abstract: Computer-based simulations are nowadays widely exploited for the predictionof the mechanical behavior of different biomedical devices. In this aspect, structural finiteelement analyses (FEA) are currently the preferred computational tool to evaluate the stentresponse under bending. This work aims at developing a computational framework basedon linear and higher order FEA to evaluate the flexibility of self-expandable carotid arterystents. In particular, numerical simulations involving large deformations and inelasticshape memory alloy constitutive modeling are performed, and the results suggest that theemployment of higher order FEA allows accurately representing the computational domainand getting a better approximation of the solution with a widely-reduced number of degreesof freedom with respect to linear FEA. Moreover, when buckling phenomena occur, higherorder FEA presents a superior capability of reproducing the nonlinear local effects related tobuckling phenomena.

J. Funct. Biomater. 2015, 6 586

Keywords: carotid artery stents; shape memory alloys; finite element analysis;stent flexibility

1. Introduction

Carotid artery stenting (CAS) is a minimally-invasive procedure widely employed for the treatmentof atherosclerosis of carotid arteries. In particular, the CAS procedure restores the physiological bloodflow by means of the expansion of a metallic endoprosthesis, i.e., the stent, which is driven to the targetlesion by means of an endoluminal path. Nowadays, CAS is considered a cost-effective alternative to thetraditional open surgery approach, leading to minimal hospitalization and reduced social and economiccosts [1,2].

Within this framework, it is important to remark that stent delivery is a complex procedure, sincethe stent needs to accommodate the tortuous path from the incision to the lesion location, avoidingstraightening the carotid artery in order to limit vessel injuries. It is immediately clear that stent designplays a crucial role in determining the mechanical properties of the device. The structural requirementsof the optimal carotid artery stent are various and, often, contradictory [3–6]. For this reason, the stentscurrently available on the market are usually the result of a trade-off among several biomechanicalfeatures, experimentally evaluated in order to find a good balance between material properties and designgeometry. However, on the one hand, experimental tests are often not applicable due to the high costsof prototype manufacturing, and on the other hand, the realistic working conditions are often difficult toreproduce experimentally.

In this aspect, modern computational methods, typically based on finite element analysis (FEA), area ubiquitous tool to simulate various clinical procedures for pre-operative planning and to predict themechanical behavior of a wide range of medical devices [7–11].

Clearly, the reliable application of such computational methods for real-life clinical and industrialproblems requires the deep comprehension of the different sources of complexity related to the problemitself. From the material point of view, the majority of modern carotid artery stents is made of NiTiNOL,a nickel-titanium shape memory alloy (SMA) able to elastically recover from strains after stress-inducedlarge deformations [12,13]. SMA behavior is inherently nonlinear, and its macroscopic properties aredriven by a complex thermomechanical solid phase transition. In this aspect, modeling such materialsrequires the development of accurate constitutive laws able to reproduce the major phenomena involvedin the material behavior, resorting to a set of parameters possibly limited in number, easy to estimateand underlying a clear physical interpretation. From the numerical point of view, the evaluation ofa particular stent feature requires a reliable transposition of the device working conditions into thecomputational model. This task is not trivial, since real-life problems often include complex nonlinearphenomena, e.g., contact or geometrical instability, that can affect the predicting capabilities of manycomputational models.

The present paper aims at investigating the capability of dedicated FEA to evaluate an importantstent feature, i.e., its flexibility (Flexibility is defined as the capability to properly bend in order


to accommodate the tortuous vascular structure, and it is considered one of the main features forcardiovascular stents), using advanced constitutive modeling, as well as realistic devices and workingconditions. Linear (h-FEA) and higher order (p-FEA) discretizations are adopted to model the 3Dstent bending problem in a large deformation regime, corresponding to the cantilever beam bendingexperiment proposed by Müller-Hülsbeck et al. [6]. Linear discretization represents a de factostandard for the numerical evaluation of stent mechanical features [14–17], while the employmentof higher order elements allows an accurate representation of the stent geometry combined withbetter approximation properties with respect to linear FEA. The simulations are performed both usingthe general purpose solver FEAP [18] and the commercial FEA software Abaqus/Standard for themost computationally-intensive simulations. The SMA model originally proposed by Souza [19] andimplemented in the version proposed by Auricchio and Petrini [20] within a large displacement-smallstrain regime is considered. The results include a performance comparison with respect to the number ofdegrees of freedom (DOF) and computational times, between h-FEA and p-FEA. Moreover, we highlightthe capability of the two methods to reproduce the nonlinear local effects due to geometrical instability.

The paper is structured as follows: In Section 2, we describe the proposed computational framework,including SMA constitutive relations, stent geometrical modeling and the implemented analysis setup.In Section 3, we present and discuss some numerical results for both h- and p-FEA. This section isstructured in order to highlight not only a general comparison in terms of the number of degrees offreedom, but also a focus on local nonlinear effects. Finally, in Section 4, we summarize our findings.

2. Materials and Methods

In this section, the SMA constitutive model used in the present study is described. In particular,the time-continuous and time-discrete frameworks, as well as the adopted large displacement-smallstrain implementation are detailed. Subsequently, the computational framework to obtain both linearand p-FEA stent models is described. The resulting model is then integrated within an analysis setupsimulating an experimental stent bending test.

2.1. Souza-Auricchio Model: Time-Continuous Framework

Following the works by Souza et al. [19] and Auricchio and Petrini [20], we adopt a 3D constitutivemodel developed within the framework of phenomenological continuum thermomechanics and able todescribe the main SMA macroscopic behaviors. The assumed control variables are the total strain εand the absolute temperature T , while the transformation strain etr is taken as an internal variable. Thetransformation strain satisfies the constraint:

‖etr‖ ≤ εL (1)

where εL is a material parameter corresponding to the maximum transformation strain reached at the endof the phase transformation during an uniaxial test.

The Helmoltz free energy density function Ψ = Ψ(ε, T, etr) is used as the thermodynamic potentialas follows:

Ψ =1

2κ θ2 +G ‖e− etr‖2 + τM ‖etr‖+

1

2h ‖etr‖2 + IεL(etr) (2)


θ and e being the volumetric and deviatoric part of the total strain ε, respectively; τM = β〈T − T ∗〉,where β is a positive parameter related to the dependence of the critical stress on temperature, T ∗

a reference temperature and 〈·〉 indicates the positive part function; κ and G are the bulk and shearmodulus, respectively; h defines the phase transformation hardening. The indicator function:

IεL(etr) =

{0 if ‖etr‖ ≤ εL

+∞ otherwise(3)

is introduced to satisfy the transformation strain constraint of Equation (1).Following standard arguments [21], the constitutive equations can be differentiated:

p =∂Ψ

∂θ= κθ

s =∂Ψ

∂e= 2G(e− etr)

X = −∂Ψ

∂etr= s− τM

etr

‖etr‖− hetr − γ

etr

‖etr‖

(4)

where p and s are the volumetric and deviatoric part of the stress σ, respectively, and X is thethermodynamic stress-like quantity associated with the transformation strain etr. The variable γ resultsfrom the indicator function subdifferential ∂IεL(etr), and it is defined as follows:

γ =

{0 if ‖etr‖ < εL

≥ 0 if ‖etr‖ = εL(5)

yielding ∂IεL(etr) = γetr/‖etr‖.A classical Mises-type limit function F = F (X) is introduced as:

F = ‖X‖ −RY (6)

where RY is a positive material parameter corresponding to the elastic radius in the deviatoric space.The evolution equation for the internal variable takes the form:

etr = λ∂F

∂X= λ

X

‖X‖(7)

where λ is the non-negative consistency parameter. The model is finally completed by the classicalKuhn–Tucker conditions:

λ ≥ 0, F ≤ 0, λF = 0 (8)

2.2. Souza-Auricchio Model: Time-Discrete Framework

Starting from the material state at time tn, identified by the quantities (etrn , λn, γn), we admit globalguess values of the total strain ε and temperature T at time tn+1. A return map procedure, based on theelastic predictor/inelastic corrector scheme [22], is adopted to compute the stress and the other variablesat the current time tn+1. For the sake of notational simplicity, we omit the subscript n + 1 for all of


the variables computed at the current time tn+1 and adopt subscript n for the variables computed at timetn. The implicit backward Euler integration scheme is employed within the evolution equation in thefollowing form:

etr = etrn + ∆λX

‖X‖(9)

where ∆λ =∫ tn+1

tnλ dt is the time-integrated consistency parameter. The algorithm consists of

evaluating an elastic trial state (denoted with subscript TR) in which the internal variables remainconstant, i.e.,

∆λTR = 0

etrTR = etrnγTR = 0

(10)

from which: sTR = 2G (e− etrTR)

XTR = sTR − τMetrTR

‖etrTR‖− hetrTR

(11)

Then, the limit function (see Equation (6)) is computed to verify the admissibility of the trial state. Ifthe trial state is admissible, the step is elastic; otherwise, the step is inelastic, and the transformationstrain has to be updated through the time-discrete evolution equation (see Equation (9)). We perform theinelastic step by solving the following non-linear system with a Newton–Raphson method: etr − etrn −∆λ

X

‖X‖= 0

‖X‖ −RY = 0

(12)

If the above solution is not admissible (i.e., if constraint (1) is not verified), a further inelasticstep is performed for saturated conditions, and the following non-linear system is solved with aNewton–Raphson method:

etr − etrn −∆λX

‖X‖= 0

‖X‖ −RY = 0

‖etr‖ − εL = 0

(13)

It is important to remark that the ratio etr/‖etr‖ is undefined when etr is null. Therefore, the euclideannorm ‖etr‖ is opportunely replaced with the following regularized expression:

‖etr‖ =√‖etr‖2 + δ −

√δ (14)

where δ is a user-defined positive regularization parameter (∼10−8). We remark that the implicitimplementation requires also the formulation of the consistent tangent matrix C (see [20] for thecorresponding expression).


2.3. Souza-Auricchio Model: Numerical Implementation

The Souza-Auricchio model is implemented as a user material subroutine (UMAT) for bothFEAP and Abaqus/Standard Version 6.11 (Dassault Systémes, Johnson, RI, USA). The hypothesisof large displacements and rotations, but small strains (as typically induced in many biomedicalapplications [17]), is assumed. In particular, starting from the deformation gradient F provided by thesolver, we compute the Green–Lagrange strain tensor as E = 1

2(FTF − I). Then, we use the following

additive decomposition of the strain tensor E:

E = e + 13θI (15)

where:θ = tr(E) e = E− 1

3θI (16)

The quantities e and θ, coupled with the temperature T and the internal variable etr, are used to

compute the second Piola–Kirchhoff stress tensor S = s + 13pI and the consistent tangent matrix C

(see Equation (4) and Section 2.2). Both S and C are expressed in terms of the reference configuration,and a push forward procedure needs to be applied in order to obtain the Cauchy stress tensor σ and theconsistent tangent matrix c in terms of the current configuration. In particular, we have:

σ = 1JFSFT (17)

c = 1Jφ∗[C] (18)

where φ∗ refers to the compact notation for the push forward operation on fourth order tensors, asdescribed in [23].

2.4. Stent Model

A novel computational framework to interface the CAD software Rhinoceros v. 4.0 SR8 (McNeeland Associated, Seattle, WA, USA) with the general purpose solver FEAP is presented. The stent modelused in the present work resembles a commercially available stent used in clinical practice, i.e., a XACTCarotid Stent (Abbott, IL, USA). This device is characterized by a closed-cell design, since all of thejunctions between different rings are connected. This feature strongly influences many biomechanicaloutcomes, e.g., vessel scaffolding, adaptability and, also, flexibility. We consider a straight configurationhaving a 9-mm reference internal diameter, a 0.18-mm thickness and a 30-mm length. Since no dataare available from the manufacturer, the main geometrical features of such devices are derived fromhigh-resolution micro-CT scans (cf. Figure 1a) of the crimped stent in the delivery system [7]. The FEAstent model is generated through the following steps:

• The geometrical features derived by the micro-CT scans are elaborated using a parametrical model,in order to obtain a geometrical description corresponding to the unfold stent in open configuration.


• A planar CAD geometry (see Figure 1b) is generated. Subsequently, a 2D CAD NURBS surfacefor the whole stent is created.

• The CAD surface structure is extruded and rolled by means of an in-house MATLAB code (TheMathWorks Inc. Natick, MA, USA) leading to the final stent in open configuration, as depicted inFigure 1c.

• The 3D trivariate CAD data are processed in order to obtain both h- and p-FEA meshes. At last,the FEA mesh is exported in a suitable format for the solver FEAP.

For the h-FEA, we employ traditional trilinear brick elements with full integration, while for thep-FEA, we employ cubic-cubic-quadratic brick elements (for circumferential, longitudinal and thicknessdirections, respectively) with full integration. In particular, the p-FEA polynomial orders are obtainedby construction of the 2D NURBS surface for circumferential and longitudinal directions, while thethickness polynomial order, linear by construction, has been raised to quadratic in order to have a fullyhigh order p-FEA mesh. Both refinement techniques are implemented using an in-house MATLAB codebased on the NURBS toolbox [24,25], a set of routines implementing the algorithms included in [26]. Inparticular, the p-FEA mesh is recovered by iterative knot insertion on a highly regular NURBS mesh.

P-FEA

FEA FEA

p-FEA

h-

Figure 1. Stent model generation: (a) detail of a high resolution micro-CT performed onthe real stents within the delivery system; (b) planar CAD geometry reproducing the stentdesign pattern; (c) 3D CAD stent model; (d) h- (top) and p- (bottom) finite element analysis(FEA) mesh generation.

2.5. Analysis Setup

Following the computational framework proposed by Auricchio et al. [27], the flexibility test issimulated through a displacement-based analysis in the large deformation regime. A displacement of11 mm along the Y direction is imposed for all of the nodes referring to the distal extremity of the stent,while the proximal one is clamped. We consider the resultant reaction force at the distal extremity ofthe device as a reference quantity to evaluate the capability of both h- and p-FEA to correctly reproducethe stent bending, as also used in the experimental setup proposed by Müller-Hülsbeck et al. [6]. Inparticular, the resultant is obtained as the sum of the reaction force contributions at the distal extremity of


the stent. The Souza-Auricchio model constitutive parameters are obtained from the literature [27]. Weuse 6 and 4 refined meshes for h- and p-FEA, respectively (the different refinement levels are shown inFigure 2). As introduced in Section 1, h-FEA-5 and h-FEA-6 simulations are computationally intensive,and thus, for efficiency reasons, they are performed using the Abaqus/Standard solver (Both h-FEAand p-FEA simulations are performed on an Intel Xeon E5-4620 at 2.20 GHz workstation with 252 GBRAM). The description of all analyses in terms of numbers of degrees of freedom (DOF) and polynomialdegrees can be found in Table 1.

p-FEA-1 p-FEA-2 p-FEA-3 p-FEA-4

h-FEA-1 h-FEA-2 h-FEA-3 h-FEA-4 h-FEA-5 h-FEA-6

p-FE

A h-

FEA

Figure 2. Stent refinement levels: top p-FEA; bottom h-FEA.

3. Results and Discussion

The present work aims at investigating the potential of dedicated FEA in simulating the stentflexibility behavior comparing the performance of h- and p-FEA discretizations. As previously indicated,we consider the resultant reaction force at the distal extremity of the stent as a reference quantity toevaluate the performance with respect to the number of DOF. In particular, the force-displacement curvesfor h-FEA and and p-FEA simulations are depicted and compared in Figure 3a,b. Moreover, reactionforce convergence plots with respect to the DOF number for both methods are reported in Figure 3,c.Finally, the data concerning reaction force values and numerical errors, evaluated with respect to theresults from the most refined p-FEA analysis (p-FEA-4) are reported in Table 1.


0 2 4 6 8 10 120

0.2

0.4

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1.6

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Forc

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0.2

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0.6

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1

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1.6

Displacement [mm]

Reac

tion

Forc

e [N

]

p−FEA 1p−FEA 2p−FEA 3p−FEA 4

0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Displacement [mm]

Reac

tion

Forc

e [N

]

IGA 1IGA 2IGA 3IGA 4

0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

Displacement [mm]

Reac

tion

Forc

e [N

]

h−FEA 6p−FEA 2

105 106 107 1080.5

1

1.5

2

DOF

Reac

tion

Forc

e [N

]

H−FEAP−FEA

105 106 107 1080.5

1

1.5

DOF

Reac

tion

Forc

e [N

]

H−FEAMIXEDP−FEA

0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

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tion

Forc

e [N

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0 2 4 6 8 10 120

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1

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0 2 4 6 8 10 120

0.2

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1

1.2

1.4

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tion

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e [N

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0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

Displacement [mm]

Reac

tion

Forc

e [N

]

h−FEA 6p−FEA 2

105 106 107 1080.5

1

1.5

2

DOF

Reac

tion

Forc

e [N

]

H−FEAP−FEA

105 106 107 1080.5

1

1.5

DOF

Reac

tion

Forc

e [N

]

H−FEAMIXEDP−FEA

0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

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1.4

1.6

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tion

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e [N

]

0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Displacement [mm]Re

actio

n Fo

rce

[N]


0 2 4 6 8 10 120

0.2

0.4

0.6

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1

1.2

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e [N

]


0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

Displacement [mm]

Reac

tion

Forc

e [N

]

h−FEA 6p−FEA 2

105 106 107 1080.5

1

1.5

2

DOF

Reac

tion

Forc

e [N

]

H−FEAP−FEA

105 106 107 1080.5

1

1.5

DOF

Reac

tion

Forc

e [N

]

H−FEAMIXEDP−FEA

0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

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1.6

Displacement [mm]

Reac

tion

Forc

e [N

]

h−FEA 1h−FEA 2h−FEA 3h−FEA 4h−FEA 5h−FEA 6

0 2 4 6 8 10 120

0.2

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0.8

1

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tion

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0 2 4 6 8 10 120

0.2

0.4

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0.8

1

Displacement [mm]

Reac

tion

Forc

e [N

]

FEA 6p−FEA 2

105 106 107 1080.5

1

1.5

2

DOF

Reac

tion

Forc

e [N

]

FEAp−FEA

105 106 107 1080.5

1

1.5

DOF

Reac

tion

Forc

e [N

]

IGAP−FEA

(a) (b) (c) Figure 3. Force-displacement diagrams for shape memory alloy (SMA) stent bending:(a) FEA; (b) p-FEA; (c) reaction force convergence plot.

Table 1. Stent bending analyses: the relative errors are evaluated with respect to the finestP-FEA simulation, labeled as P-FEA-4. The symbols p, q and r indicate the circumferential,longitudinal and thickness polynomial orders of the FEA meshes, respectively.

Mesh Label DOF Order Reaction Force (N) Critical Load (N)

p q r Value Error Value Error

FEA-1 606,276 1 1 1 1.0102 60.21% 1.3354 47.76 %FEA-2 1,635,960 1 1 1 0.91935 51.85% 1.1671 29.14 %FEA-3 2,118,096 1 1 1 0.89916 42.60 % 1.1300 25.03 %FEA-4 3,246,480 1 1 1 0.79611 26.25 % 1.0754 18.99 %FEA-5 5,281,740 1 1 1 0.73349 16.32 % 0.99991 10.64 %FEA-6 10,622,016 1 1 1 0.68897 9.26 % 0.97022 7.35 %

p-FEA-1 598,212 3 3 2 0.7725 22.51 % 1.0342 14.31 %p-FEA-2 1,844,820 3 3 2 0.6732 6.76% 0.9544 5.49 %p-FEA-3 3,469,668 3 3 2 0.6480 2.76 % 0.91242 0.8 %p-FEA-4 5,269,642 3 3 2 0.63054 – 0.90473 –

The results show that, as expected, p-FEA presents a better performance with respect to h-FEA on aper degree of freedom basis, with a gain of about one order of magnitude in terms of DOF number. Inparticular, except the coarsest mesh p-FEA-1, which presents some unphysical local buckling, all of thep-FEA meshes show a numerical error below 7%. As an example, we fix the DOF number in the rangeof 2 · 106, corresponding to the meshes p-FEA-2 and h-FEA-3, respectively. In particular, p-FEA-2shows a numerical error of 6.76% with respect to the converged solution p-FEA-4, while h-FEA-3shows a relative error of 42.60%. Moreover, the most refined h-FEA, comprising over ten million DOF,still shows an error of 9.26%.

As introduced previously, we now focus on the influence of stent design, with particular care for kinkformation and the buckling phenomenon, which often appear when a closed-cell stent is considered.Kink resistance is an important feature of stent devices [28]. When strains locally increase beyond the


critical value, the buckling phenomenon occurs, inducing high stresses. This phenomenon can be verydangerous for device performance, since it can lead to the reduction of the fatigue life and implant failure.In particular, closed-cell designs show reduced adaptability and are prone to kinking [7].

Our results confirm this statement, also in accordance with experimental results [6] (Figure 4b,c).From a computational viewpoint, in Figure 3, it is possible to observe that, while p-FEA presentsthe same deformation pattern for all considered refinements, h-FEA shows different behaviors withdifferent refinements, which are related to some spurious stress concentrations that lead to an erroneousreproduction of the buckling deformation path. In particular, the p-FEA deformation pattern shows twostages of local buckling regardless of the refinement level (see Figure 3b). On the other hand, h-FEAis not able to catch this local behavior until a high number of DOF is included (see Figure 3a). Thisphenomenon can be better appreciated in Figure 4a. This aspect has a great influence on the capabilityof accurately reproducing the value of the critical load (see Table 1).

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FEAP−FEA

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IGAP−FEA

(a) (b) (c)

P-FEA 2 FEA 8

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FEA 6p−FEA 2

105 106 107 1080.5

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1.5

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DOF

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FEAp−FEA

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IGAP−FEA

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P−FEA 1P−FEA 2P−FEA 3P−FEA 4

0 2 4 6 8 10 120

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0.4

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1

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tion

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e [N

]

FEA 8P−FEA 2

104 105 106 107 1080.5

1

1.5

2

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DOF

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tion

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e [N

]

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104 105 106 107 1080.5

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tion

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e [N

]

IGAP−FEA

(a) (b) (c)

P-FEA 2 FEA 8 p-FEA 2 FEA 6

0 2 4 6 8 10 120

0.2

0.4

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1

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tion

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e [N

]

0 2 4 6 8 10 120

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1

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Displacement [mm]

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tion

Forc

e [N

]


0 2 4 6 8 10 120

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1

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]


0 2 4 6 8 10 120

0.2

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h−FEA 6p−FEA 2

105 106 107 1080.5

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tion

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e [N

]

H−FEAP−FEA

105 106 107 1080.5

1

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tion

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e [N

]

H−FEAMIXEDP−FEA

Figure 4. FEA versus p-FEA comparison: (a) force displacement curves obtained usingthe finest FEA mesh (FEA-8) and a coarse p-FEA mesh (p-FEA-2); (b) p-FEA-2 deformedconfiguration; (c) FEA-8 deformed configuration.

3.1. Computational Times

Even though a fair efficiency comparison of h-FEA and p-FEA stent bending simulations requires theemployment of the same software package, it is interesting to provide the computational times for a givenlevel of accuracy. In particular, the computational time coming from two simulations with comparablelevels of accuracy, i.e., p-FEA-2 mesh and h-FEA-6 mesh, are reported in Table 2. In this case, while thep-FEA-2 simulation is performed with FEAP and one CPU, the h-FEA-6 simulation is performed withAbaqus/Standard with eight CPUs, for the reason of the motivations previously mentioned. However,it is interesting to remark that, from a qualitative point of view, p-FEA shows comparable times withrespect to h-FEA, despite the difference in the number of used CPUs.

Table 2. Computational times for p-FEA and h-FEA.

Mesh Label DOF No. of CPUs Solver Total Analysis Time

p-FEA-2 1,844,820 1 FEAP 27 h 15 minFEA-6 10,622,016 8 Abaqus/Standard 26 h 23 min


4. Conclusions

In the present study, we developed a numerical bench test able to integrate stent FEA models andinelastic SMA constitutive modeling, in order to evaluate the flexibility of self-expandable carotid arterystents. We implemented a computational framework able to employ both h-FEA and p-FEA, with theultimate goal to compare the numerical performance of high order basis with respect to the standardlinear basis. Our results suggest that the employment of p-FEA allows one to accurately represent thecomputational domain and to obtain a better approximation of the solution with a reduced number ofDOF with respect to h-FEA. Moreover, when buckling phenomena occur and geometrical accuracy playsan important role, p-FEA presents a superior capability to reproduce the nonlinear local effects. The fullexploitation of the present framework within the stent development requires different advancements,including the coupling of flexibility test with imaging techniques of real devices, the calibration of theSMA model parameters, the introduction of realistic vessel models and the implementation of a robustcontact driver to simulate the stent-vessel interaction. Moreover, the cutting-edge research on stentsrequires the use of explicit dynamics solvers, able to get rid of the high nonlinearity of such simulations.In this aspect, it is well known in the literature that p-FEA are not able to correctly reproduce a wide rangeof the frequency spectrum, while the use of linear elements mitigate this adverse effect [29,30]. Somerecent results [31] demonstrate that the introduction of more sophisticated basis, i.e., NURBS-basedisogeometric analysis [32], characterized by high order and high regularity basis functions, providesexcellent results also for explicit dynamics simulations [33].

Acknowledgments

This work is funded by the European Research Council Starting Grant through the Project ISOBIONo. 259229, by the Cariplo Foundation through the Project iCardioCloud No. 2013-1179 and byRegione Lombardia through the Project No. E18F13000030007.

Author Contributions

M.F, A.R., and F.A. conceived the approach; E.B., G.S., and F.A. implemented the SMA constitutivemodel; M.F. performed the numerical experiments; M.F, A.R., M.C., and S.M. analyzed the data; M.F.wrote the paper; M.F., A.R., M.C., and S.M. revised the paper.

Conflicts of Interest

The authors declare no conflict of interest.

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c© 2015 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access articledistributed under the terms and conditions of the Creative Commons Attribution license(http://creativecommons.org/licenses/by/4.0/).

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